Method for evaluating error in shape of free curved surface

ABSTRACT

An original curved surface S is divided into up to six curved surface units by combinations of signs (+, 0, −) of a principal curvature (k 1 , k 2 ) in each point on the curved surface. A distorted curved surface S′ is associated with the original curved surface S and divided into curved surface units having the same boundary. An average normal vector is obtained for each curved surface unit with respect to the original curved surface and the distorted curved surface. “A bent component” and “a twisted component” in all the combinations of pairs of different curved surface units are obtained with respect to the original curved surface and the distorted curved surface. A difference between “the bent component” and “the twisted component” in each combination in the original curved surface and “the bent component” and “the twisted component” in each identical combination in the distorted curved surface is calculated.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a method for evaluating a shape errorof a free curved surface.

2. Description of the Prior Art

Conventionally, in press working or injection molding processing, acurved surface (i.e. an original curved surface), such as target CAD(Computer Aided Design) data and a curved surface associated therewith(i.e. a distorted curved surface), are typically “visually” compared.Any forming defects in the working or the processing are evaluated basedon a difference, noted during the “visual” inspection, between theseoriginal and distorted curves. That is, for example in press working, ifthe distorted curved surface has been “bent” and “twisted” relative tothe original curved surface, both curved surfaces are visually judgedagainst one another and the metal mold is experientially adjusted.

The above-described visual experiential evaluation of the curves issimple and convenient, but it has problems to include large individualdifferences, dependence on the evaluator's experience, and extremelyhigh arbitrariness. Therefore, a means for evaluating the entire bendsand twists of the free curved surface by an objective, less arbitrarymeans for evaluation is desirable.

On the other hand, for this purpose, a formed article (i.e., distortedcurved surface) that was actually formed by using, for example, formedsheet metal and the like, is measured by a Coordinate Measuring Machineor a digitizer. In this way, an image of the obtained measured result isdisplayed together with the original curved surface (i.e., CAD data andothers) so that a shape error, such as bent or twisted shape error, canbe roughly recognized by visual inspection. With this technique,however, it is difficult to visually recognize three-dimensionaldifferences. Specifically, when a reference shape is not flat but has acomplicated curve, a difference in the three-dimensional measurementbetween the distorted curved surface and the original curved surface canhardly be visually recognized.

Furthermore, if the distorted curved surface has a local “wrinkle”,“bump” or “dent”, the image is largely changed and the entire shapeerror is hard to evaluate by means of the conventional method.

SUMMARY OF THE INVENTION

The present invention is intended to solve the above-mentioned problems.Thus, a main object of the present invention is to provide a method forevaluating a shape error of a free curved surface, wherein an entireshape of an original curved surface, such as CAD data, is compared withan entire shape of a distorted curved surface, such as provided afterforming, to easily and objectively determine a difference (i.e., error).Another object of the present invention is to provide a method capableof evaluating the entire shape of a free curved surface without beingaffected by a local “wrinkle”, “bump” or “dent” in the free curvedsurface. Yet another object of the present invention is to provide amethod for evaluating a shape error of a free curved surface in whichthe numerical calculation is facilitated, and the influence of errors inthe numeric values and in the measurements are minimized so the methodcan be applied to both a parametric curved surface and a cloud ofpoints.

According to the present invention, a method for evaluating a shapeerror of a free curved surface is provided which comprises: (a) step Afor dividing an original curved surface S into up to six curved surfaceunits by combinations of signs (+, 0, −) of a principal curvature (K₁,K₂) in each point on the curved surface; (b) step B for associating adistorted surface S′ with the original curved surface S and dividing thedistorted curved surface S′ into curved surface units having the sameboundary by projection along the normal vectors of S; (c) step C forobtaining an average normal vector for each curved surface unit withrespect to the original curved surface and the distorted curved surface;(d) step D for obtaining “a bent component” and “a twisted component” ofall combinations of pairs of the different curved surface units withrespect to the original curved surface and the distorted curved surface;and (e) step E for calculating a difference between “a bent component”and “a twisted component” of the respective components in the originalcurved surface and “a bent component” and “a twisted component” of thesame respective components in the distorted curved surface.

According to the above method of the present invention, if the curvedsurface remains to be continuous even after distortion, localirregularities (i.e., wrinkles, bumps or dents) can be canceled bytaking an average of the normal vectors in each region (i.e., curvedsurface unit) of the free curved surface, and a global direction of thatregion (i.e., average normal vector) can be determined. Therefore,geometrical properties (i.e., a bent component, and a twisted component)of the curved surface can be readily and objectively evaluated, withoutambiguity, based on the directional relationship relative to anotherregion (i.e., another curved surface unit) of the average normal vector.

Moreover, since the geometrical properties (i.e., a bent component, anda twisted component) can be easily calculated using only the normalvector in each point on the curved surface, the numerical calculation isfacilitated, and the influence of numeric errors and measurement errorsis reduced. Consequently, the method in accordance with the presentinvention can be applied to both a parametric curved surface and cloudsof points.

Furthermore, according to a preferred embodiment of the presentinvention, each curved surface unit is divided into two directionsorthogonal to each other and the average normal vectors of the twodivided regions are calculated in step C, and “a bent component” and “atwisted component” are obtained with respect to combinations of pairs ofthe average normal vectors in the two divided regions in the respectivecurved surface units in step D.

With this method in accordance with the present invention, dividing thesame curved surface units in the original curved surface S into twodirections provides two pairs of the bent component and the twistedcomponent in the corresponding region, as well as the above-mentionedrelationship relative to another region. Thus, it is possible toevaluate both the relationship with another region, and a change inshape in the corresponding region, with respect to each curved surfaceunit by this two-stage evaluation. In this way, the two-stage evaluationperforms a more accurate shape error evaluation.

Other objects and advantageous characteristics of the present inventionwill be apparent from the following description with reference to theaccompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a flow chart outlining a method according to the presentinvention.

FIGS. 2A, 2B and 2C are schematic drawings corresponding to the flowchart of FIG. 1.

FIG. 3 shows the relationship between two normal vectors, a bentcomponent θ and a twisted component φ.

BRIEF DESCRIPTION OF THE PREFERRED EMBODIMENTS

A free curved surface S=S(u, v) can be expressed by parameters u and v.Expression 3 is a relational expression in the differential geometry.This relationship is disclosed in, for example, “Curves and Surfaces forComputer Aided Geometric Design” (Farin, G, 1988, A Practical Guide.Academic Press).

Expression 3 is as follows:

E=S _(M) S _(M) ,F=S _(M) S _(V) ,G=S _(V) S _(V) ,L=nS _(MM),

$\begin{matrix}{{M = {nS}_{uv}},{N = {nS}_{w}},{n = \frac{S_{u} \times S_{v}}{{}S_{u} \times S_{v}{}}}} & (1)\end{matrix}$

The principal curvature is K₁, K₂, which can be obtained by solvingFormula 2 of Expression 4.

Expression 4 is as follows: $\begin{matrix}{\kappa^{2} = {{{\frac{{NE} - {2{MF}} + {LG}}{{EG} - F^{2}}\quad \kappa} + \frac{{LN} - M^{2}}{{EG} - F^{2}}} = 0}} & (2)\end{matrix}$

The method according to the present invention will now be described.FIG. 1 is a flow chart typically showing the method according to thepresent invention, and FIGS. 2A, 2B and 2C are schematic drawingscorresponding to this flow chart. As shown in these drawings, the methodaccording to the present invention basically comprises five steps A toE.

In a first step A, an original curved surface S is divided into up tosix curved surface units by combinations of signs (+, 0, −) of theprincipal curvature (K₁, K₂) in each point on the original curvedsurface. The more the number of points for calculating the principalcurvature increases, the more the accuracy can be improved.Additionally, the six possible combinations of the signs (+, 0, −) ofthe principal curvature (K₁, K₂) are: (+, +); (+, −); (+, 0); (−, −);(−, 0); and (0, 0). For example, (+, +) describes a convex surface suchthat both principal curvatures (K₁, K₂) are convex; (−, −) describes aconcave surface such that both principal curvatures are concave; (+, −)describes a saddle-like surface such that both principal curvatures areconcavo-convex; and (+, 0), (−, 0) or (0, 0), each describes a curvedsurface (cylindrical surface and the like) such that one of the twoprincipal curvatures is linear. Therefore, the original curved surface Scan be divided into up to six curved surface units (e.g., two curvedsurface units 1 and 2 in the example shown in FIGS. 2A, 2B and 2C)having the combined shape approximating, by dividing into curved surfaceunits in step A, the original curved surface S. It is noted that thedividing method using combinations of the principal curvature s isdisclosed in, for example, CAD Jrnl. V.30, N6. pp473-486(1998), R. Gadhand R. Sonthi: “Geometric shape abstractions for internet-based virturalprototyping,” and by others.

In a second step B, a distorted curved surface S′ is associated with theoriginal curved surface S, and also divided into the same number ofcurved surface units having the same boundary by projection along normalvectors of S, as shown in FIGS. 2A and 2B. In this case, each curvedsurface unit, for example 1 and 2, of the distorted curved surface S′may include different combinations of the principal curvature s.

In a third step C, the average normal vector is calculated for eachcurved surface unit, for example 1 and 2, with respect to the originalcurved surface S and the distorted curved surface S′ as shown in FIG.2C. Additionally, in the third step, each curved surface unit is dividedinto two directions orthogonal to each other. For example, as shown inFIG. 2C, the two directions orthogonal to each other could be 1 a, 1 band 2 a, 2 b, or 1 c, 1 d and 2 c, 2 d. Subsequently, the normal averageline vectors in each of the two divided regions of each curved surfaceunit are obtained. The method for calculating the average normal vectorwill be described later.

In a fourth step D, “a bent component” and “a twisted component” in allthe combinations of the different curved surface units are obtained bycalculation with respect to the original curved surface S and thedistorted curved surface S′ from the later-described formula.Furthermore, in step D, for each curved surface unit, “a bent component”and “a twisted component” in the combinations of pairs of the averagenormal vectors in the two divided regions are obtained.

A fifth step E calculates a difference between “a bent component” and “atwisted component” in each combination in the original curved surfaceand “a bent component” and “a twisted component” in each of the samecombinations in the distorted curved surface S′.

If a plurality of curved surface units are defined in the originalcurved surface S in the above-described fourth step D, “a bentcomponent” and “a twisted component” are obtained with respect to allthe combinations of pairs of different curved surface units.Furthermore, as to the distorted curved surface S′, “a bent component”and “a twisted component” are likewise obtained with respect to thecorresponding curved surface units. Accordingly, when a difference of “abent component” and “a twisted component” of combinations of thecorresponding curved surface units is calculated in the fifth step E,the geometrical properties (i.e., a bent component, a twisted component,and a shape error as a difference between these two components) of thedistorted curved surface S′ relative to the original curved surface Scan be objectively evaluated.

Furthermore, a change in shape in each curved surface unit can beobjectively evaluated by (i) dividing each curved surface unit into tworegions corresponding to the two orthogonal directions and obtaining theaverage normal vectors in the two divided regions in the third step Cand by (ii) obtaining “a bent component” and “a twisted component” withrespect to the combinations of pairs of the average normal vectors inthe two divided regions in the fourth step D. Therefore, both therelationship relative to another region, and a change in shape in thecorresponding region, can be evaluated with respect to each curvedsurface unit by performing this two-stage evaluation, thereby providinga more accurate shape error evaluation.

The calculation technique in the above-described method will now bedescribed.

The average normal vector in the third step C corresponds to a result ofa division of a synthetic vector, which is the sum of n unit normalvectors in respective points (n points: intersection points of normalvectors in the respective points on the original curved surface andstraight lines parallel thereto in the case where the curved surface isa distorted curved surface), on the curved surface by n. In this case,it is determined that the starting points of the average normal vectorsare the same in both the original curved surface S and the distortedcurved surface S′, and an average (i.e., a result of division of thesynthetic vector of positional vectors at the respective points by n) ofn points used for calculating the unit normal vector in the originalcurved surface is used. Therefore, one normal vector is defined for eachcurved surface unit.

In addition, the normal vector for each point on the curved surfacecorresponds to n in Formula (1) when the free curved surface isexpressed as a parametric curved surface S=S(u, v). Moreover, when thecurved surface is represented as several point groups on that curvedsurface, the normal vector can be approximately obtained by calculatingan outer product of vectors toward adjacent two points which are not onthe same straight line in each point. Thereafter, the average can beobtained after normalization (also referred to as “unit vectorization”),as similar to the case of the parametric curved surface. Therefore, themethod according to the present invention can be similarly applied tothe parametric curved surface and the point group.

Furthermore, when the surface remains to be a continuous curved surfaceafter distortion, the local irregularities (i.e., wrinkles, bump or dentand others) can be canceled. Even if wrinkles or surface sinks exist onthe distorted curved surface S′, they can be offset so that the overallshape of the curved surface unit can be expressed by a single syntheticvector (i.e., average normal vector).

FIG. 3 shows a technique for obtaining a bent component θ and a twistedcomponent φ from the two normal vectors representative of the curvedsurface unit. As shown in the FIG. 3, assuming that: vectors AB and CDare average normal vectors with the respective points A and C asstarting points; a vector CD′, which is an orthogonal projection fromthe vector CD to a plain surface defined by three points A, B and C; anda vector AD″, which is a vector parallel to the vector CD′, it ispossible to define that the bent component θ is an angle formed betweenthe vectors AB and AD″ and a twisted component φ is an angle formed by aplain surface ABC and a counterpart surface ACD.

Formulas (3) to (7) of Expressions 5 and 6 represent the bent componentand the twisted component in the form of formulas. A sign of the bentcomponent θ is determined by a function “sgn” as shown by Formula (3).If the sign of the element in parentheses of the function sgn(.) ispositive, 1.0 is returned, and if the element is negative, −1.0 isreturned. The normal vector of the plain surface ABC is determined as avector n calculated by Formula (4), and Formula (5) is used to express aprojection of the vector CD to the plain surface ABC as an angle with asign as shown in Formula (6). It is to be noted that Arc cos denotes aninverse function of cosine.

Expression 5 $\begin{matrix}{{{sgn}\left( {\overset{\rightarrow}{AB},\overset{\rightarrow}{CD},\overset{\rightarrow}{CA}} \right)} = \left\{ {\begin{matrix}{+ 1} & {{{if}\quad {\left( {\overset{\rightarrow}{AB} - \overset{\rightarrow}{CD}} \right) \cdot \overset{\rightarrow}{CA}}} > 0} \\{- 1} & {else}\end{matrix},} \right.} & (3) \\{\overset{\rightarrow}{n} = \frac{\overset{\rightarrow}{AB} \times \overset{\rightarrow}{AC}}{{\overset{\rightarrow}{AB} \times \overset{\rightarrow}{AC}}}} & (4) \\{\overset{\rightarrow}{{CD}^{\prime}} = {\overset{\rightarrow}{CD} - {\left( {\overset{\rightarrow}{n} \cdot \overset{\rightarrow}{CD}} \right)\overset{\rightarrow}{n}}}} & (5) \\{{\theta = {{{sgn}\left( {\overset{\rightarrow}{AB},\overset{\rightarrow}{CD},\overset{\rightarrow}{CA}} \right)}{{{Arccos}\left( \frac{\overset{\rightarrow}{AB} \cdot \overset{\rightarrow}{{CD}^{\prime}}}{{\overset{\rightarrow}{AB} \cdot \overset{\rightarrow}{{CD}^{\prime}}}} \right)}}}},} & (6)\end{matrix}$

On the other hand, the twisted component φ can be expressed as an anglewith a sign function as shown by Formula (7). In Formula (7), a signrepresentative of a direction is added to an angle formed between theplain surface ABC and the counterpart ACD with AC as a nodal line. It isto be noted that Arc sin is an inverse function of sin in Formula (7).

Expression 6 $\begin{matrix}{\varphi = {{Arcsin}\left( {\frac{\left( {\overset{\rightarrow}{AC} \times \left( {\overset{\rightarrow}{AB} \times \overset{\rightarrow}{AC}} \right)} \right) \times \left( {\overset{\rightarrow}{AC} \times \left( {\overset{\rightarrow}{CD} \times \overset{\rightarrow}{AC}} \right)} \right)}{{\left( {\overset{\rightarrow}{AC} \times \left( {\overset{\rightarrow}{AB} \times \overset{\rightarrow}{AC}} \right)} \right){}\left( {\overset{\rightarrow}{AC} \times \left( {\overset{\rightarrow}{CD} \times \overset{\rightarrow}{AC}} \right)} \right)}} \cdot \frac{\overset{\rightarrow}{AC}}{\overset{\rightarrow}{AC}}} \right)}} & (7)\end{matrix}$

where {right arrow over (AB)}×{right arrow over (CD)} represents anouter product of the vectors of AB and CD.

As described above, the method for evaluating a shape error of a freecurved surface according to the present invention is advantageous inthat: (a) the original curved surface, such as CAD data, can be readilyand objectively compared with the distorted curved surface after formingfor the entire shape, and a difference (error) thereof can bedetermined; (b) the entire shape of the free curved surface can beevaluated without being affected by local “wrinkles”, a “bump” or a“dent”; (c) the numerical calculation is facilitated; (d) the influenceof numeric error and measurement error is reduced; and (e) the methodcan be applied to both a parametric curved surface and a point group.

Although the present invention has been explained based on the preferredembodiments, it is understood that the scope of the invention is notrestricted to these embodiments. On the contrary, the true scope of theinvention includes improvements, modifications and equivalents withinthe appended claims.

What is claimed is:
 1. A method for evaluating a shape error of a freecurved surface comprising: dividing an original curved surface S into upto six curved surface units by combinations of signs (+, 0, −) of aprincipal curvature (k₁, k₂) in each point on said curved surface;associating a distorted curved surface S′ with said original curvedsurface S and dividing it into curved surface units having the sameboundary by projection along normal vectors of the curved surface S;calculating an average normal vector for each curved surface unit withrespect to said original curved surface and said distorted curvedsurface; obtaining “a bent component” and “a twisted component” in allthe combinations of pairs of different curved surface units with respectto said original curved surface and said distorted curved surface; andcalculating a difference between “the bent component” and “the twistedcomponent” in each combination in said original curved surface and “thebent component” and “the twisted component” in each identicalcombination in said distorted curved surface.
 2. A method for evaluatinga shape error of a free curved surface comprising the steps of: dividingan original curved surface S into up to six curved surface units bycombinations of signs (+, 0, −) of a principal curvature (K₁, K₂) ineach point on the curved surface S; associating a distorted curvedsurface S′ with the original curved surface S and dividing the distortedcurved surface S′ into curved surface units having the same boundary byprojection along normal vectors of the curved surface S; calculating anaverage normal vector for each curved surface unit with respect to theoriginal curved surface S and the distorted curved surface S′, whereineach curved surface unit is divided into two regions having orthogonaldirections and average normal vectors in the two divided regions arecalculated; obtaining a bent component and a twisted component in allcombinations of pairs of different curved surface units with respect tothe original curved surface S and the distorted curved surface S′,wherein the bent component and the twisted component are obtained withrespect to combinations of pairs of the average normal vectors in thetwo divided regions in each curved surface unit; and calculating adifference between the bent component and the twisted component in eachcombination in the original curved surface S and the bent component andthe twisted component in each identical combination in the distortedcurved surface S′.
 3. A method for evaluating a shape error of a freecurved surface according to claim 2, wherein a bent component θ iscalculated by Expression 1: Expression 1 $\begin{matrix}{{{sgn}\left( {\overset{\rightarrow}{AB},\overset{\rightarrow}{CD},\overset{\rightarrow}{CA}} \right)} = \left\{ {\begin{matrix}{+ 1} & {{{if}\quad {\left( {\overset{\rightarrow}{AB} - \overset{\rightarrow}{CD}} \right) \cdot \overset{\rightarrow}{CA}}} > 0} \\{- 1} & {else}\end{matrix},} \right.} & (3) \\{\overset{\rightarrow}{n} = \frac{\overset{\rightarrow}{AB} \times \overset{\rightarrow}{AC}}{{\overset{\rightarrow}{AB} \times \overset{\rightarrow}{AC}}}} & (4) \\{\overset{\rightarrow}{{CD}^{\prime}} = {\overset{\rightarrow}{CD} - {\left( {\overset{\rightarrow}{n} \cdot \overset{\rightarrow}{CD}} \right)\overset{\rightarrow}{n}}}} & (5) \\{{\theta = {{{sgn}\left( {\overset{\rightarrow}{AB},\overset{\rightarrow}{CD},\overset{\rightarrow}{CA}} \right)}{{{Arccos}\left( \frac{\overset{\rightarrow}{AB} \cdot \overset{\rightarrow}{{CD}^{\prime}}}{{\overset{\rightarrow}{AB} \cdot \overset{\rightarrow}{{CD}^{\prime}}}} \right)}}}},} & (6)\end{matrix}$

and a twisted component φ can be calculated by Expression 2: Expression2 $\begin{matrix}{\varphi = {{Arcsin}\left( {\frac{\left( {\overset{\rightarrow}{AC} \times \left( {\overset{\rightarrow}{AB} \times \overset{\rightarrow}{AC}} \right)} \right) \times \left( {\overset{\rightarrow}{AC} \times \left( {\overset{\rightarrow}{CD} \times \overset{\rightarrow}{AC}} \right)} \right)}{{\left( {\overset{\rightarrow}{AC} \times \left( {\overset{\rightarrow}{AB} \times \overset{\rightarrow}{AC}} \right)} \right){}\left( {\overset{\rightarrow}{AC} \times \left( {\overset{\rightarrow}{CD} \times \overset{\rightarrow}{AC}} \right)} \right)}} \cdot \frac{\overset{\rightarrow}{AC}}{\overset{\rightarrow}{AC}}} \right)}} & (7)\end{matrix}$

where {right arrow over (AB)}×{right arrow over (CD)} represents anouter product of the vectors of AB and CD.